MATHEMATICAL SECTION. TA 
plementary nomes at the several steps of the modular scale are as 
follows 
—n Die Gist 
m= o ee B= 3H=C GT=RIEHTY 
SSGon) ss ote qQ”) = (71) 
Qn he 2? ay Ee (Aaeee 2 Ud 
Pri=p ape > 1—P-s P—P> Pp —P>P 
9-2 Q-n 
Sar OIC Arata (ets op” = p (72) 
We have then in this transformation the simplest possible case 
of Abel’s theorem (27); and because in ascending we pass to the 
square of the nome, it is called the quadric transformation. 
The ascending transformation is possible in real quantities if 
c¢ > a, for we have M(c, a) = M(a,c). Also if 6 > awe can use 
the descending transformation; and in either case we can, after 
one transformation, proceed in either direction. This may be 
symbolized by the following diagram 
bg a’ > rd 
ae 
In order to exhibit the practical nature of the formule given, I 
shall make the necessary aca area for the integral 
fe ae 
aa V1—sin?75sin’¢ Q 
if ¢ = 70° and also for tho complete integral. . 
Because y = sin 75° is > 7/3 we must use the ascending trans- 
formation. The computation for 70°sin 7° may be conveniently 
arranged as follows: 
26 
